FCC-ee: coupling impedances and collective effects E.Belli M.Migliorati, G.Rumolo Acknowledgements: M. Zobov, G.Castorina, B.Spataro, S. Persichelli Impedance meeting May 12, 2017

Outline Resistive wall (RW) Single bunch effects Microwave Instability (MI) Transverse mode coupling instability (TMCI) Multibunch effects Other impedance sources Conclusions

FCC-ee parameter list for instabilities studies K. Oide

+ Resistive wall Vacuum chamber with 35mm radius Non Evaporable Getter (NEG) thk =1𝜇m 𝜌= 10 −6 Ωm Best choice for vacuum (low SEY, low desorption yield) MI threshold below nominal intensity Three layers (No Coating) Cu 2mm 𝜌=1.66∙ 10 −8 Ωm Dielectric 6mm 𝜌= 10 15 Ωm Stainless Steel Infinity 𝜌= 6.89∙10 −7 Ωm Titanium Nitride (TiN) thk = 200nm 𝜌=0.5∙ 10 −6 Ωm + Low SEY, thinner coating Vacuum pumps for desorption Amorphous Carbon (AC) thk = 200 nm 𝜌= 10 −4 Ωm Low SEY, thinner coating, low desorption yield

RW - Transverse and longitudinal impedances ImpedanceWake2D ImpedanceWake2D

RW – Single bunch effects Main single bunch effects of the RW on the single bunch dynamics Microwave Instability in the longitudinal plane Transverse Mode Coupling Instability in the transverse plane Bunch intensity limitation Frequencies of the intra-bunch modes ( Ω=𝑚 𝜔 𝛽 +𝑙 𝜔 𝑠 ) depend on the bunch intensity As the bunch intensity increases, the mode frequencies shift and merge, giving rise to the instability Instability threshold localized in the point where there is mode coupling. Above the threshold: Transverse case Bunch loss Longitudinal case Increase of the bunch length and energy spread Bunch internal oscillations (dangerous in beam-beam collisions) DELPHI

MI - Longitudinal impedance Higher values of impedance for NEG Im[ 𝑍 𝑇𝑖𝑁 ]≈Im[ Z AC ] Re[ 𝑍 𝑁𝐶 ]≈Im[ Z NC ]≈Re[ 𝑍 𝑇𝑖𝑁,𝐴𝐶 ]

MI – Longitudinal wake potential Macroparticle simulations (PyHEADTAIL, SBSC) Wake potential of a bunch as the convolution between the wake function, i.e. the wake potential of a point charge, and the longitudinal bunch distribution Variation of the wake function in short distances  Simulations very time consuming due to the large number of slices needed Wake potential of a very short bunch as Green function for the code 𝝈 𝒛 =𝟎.𝟐𝟏𝒎𝒎 𝑊 ∥ 𝑧 = −∞ 𝑧 𝜔 ∥ 𝑧− 𝑧 ′ 𝜆 𝑧 ′ 𝑑𝑧′

MI – Haissinski equation The Haissinski equation gives the bunch length and the distortion of the Gaussian distribution in the presence of wake fields and RF external voltage for intensities below the MI threshold 𝜆 𝑧 = 𝜆 0 𝑒 1 𝐸 0 𝜂 𝜎 𝑑𝑝 2 Ψ(𝑧) with Ψ z = 1 𝐶 0 𝑧 𝑒 𝑉 𝑅𝐹 𝑧 ′ − 𝑈 0 𝑑 𝑧 ′ − 𝑒 2 𝑁 𝑝 𝐶 0 𝑧 𝑑𝑧′ −∞ 𝑧′ 𝜆 𝑧 ′′ 𝜔 ∥ 𝑧− 𝑧 ′′ 𝑑𝑧′′ 𝝈 𝒛 =𝟐.𝟏𝒎𝒎 𝝈 𝒛 =𝟐.𝟒𝒎𝒎 𝝈 𝒛 =𝟐.𝟔𝒎𝒎

MI – Bunch length and energy spread 60 ∘ optics (45.6 GeV), No Beamstrahlung ( 𝜎 𝑧 =2.1𝑚𝑚) MI instability threshold ≃1.6 −1.8 ⋅ 10 11 in the case of no coating Around the nominal bunch intensity in the case of TiN and AC Below the nominal bunch intensity in the case of NEG MI regime

MI – Bunch length and energy spread (WBS) Beamstrahlung, i.e. SR during the collision with the opposite beam Total energy spread Total bunch length 𝜎 𝑑𝑝,𝑡𝑜𝑡 = 1 2 𝜎 𝑑𝑝,𝑆𝑅 2 + 1 4 𝜎 𝑑𝑝,𝑆𝑅 4 +𝐴 𝜎 𝑑𝑝,𝑆𝑅 2 𝜎 𝑧,𝑆𝑅 2 =𝟎.𝟏𝟏𝟐% 𝐴=1.4 𝑛 𝐼𝑃 𝜏 𝐸,𝑆𝑅 4 𝑇 0 𝑟 𝑒 5 𝑁 𝑏 3 𝛾 2 𝛼 𝜎 𝑥 3 with 𝜎 𝑧,𝑡𝑜𝑡 = 𝛼 𝑐 𝐶 2𝜋 𝑄 𝑠 𝜎 𝑑𝑝,𝑡𝑜𝑡 =𝟔.𝟐𝟐𝒎𝒎 Stable beams

MI – Coating thickness? 60° optics (45.6 GeV), No BS ( 𝜎 𝑧 =2.1𝑚𝑚) Does the material thickness affect the MI? Example: AC The MI instability threshold is 3x higher in the case of 25nm thickness

MI – Increasing 𝜎 𝑧 and 𝑁 𝑝 D. Shatilov, “Recent results about the X-Z instability”, 53th FCC-ee Optics Design meeting in the same proportion Boussard criterion to obtain a first estimation of the MI threshold 𝑍 ∥ 𝑛 𝑛 ≤ 2 𝜋 3 2 𝐸 0 𝑐 𝑒 2 𝛼 𝑐 𝜎 𝑧 𝑁 𝑝 𝜎 𝑑𝑝 2

TMCI seems to be not dangerous TMCI simulations 60 ∘ optics (45.6 GeV), No BS ( 𝜎 𝑧 =2.1𝑚𝑚) Macroparticle simulations with PyHEADTAIL Analytic simulations with DELPHI (N.Mounet), by including the bunch lengthening due to the longitudinal wake TMCI seems to be not dangerous

RW - Transverse coupled bunch instability Growth rate for the 𝜇-th mode 𝟏 𝝉 𝟕𝟑𝟓𝟎𝟒 =𝟒𝟗𝟒.𝟖 𝒔 −𝟏 ≃𝟔 𝒕𝒖𝒓𝒏𝒔   =1 with     𝑅𝑒 [ 𝑍 ⊥ 𝜔 ]=𝑠𝑔𝑛(𝜔) 𝐶 2𝜋 𝑏 3 2 𝑍 0 𝑐 𝜎 𝑐 |𝜔| Negative 𝜔  unstable mode with exponential growth Positive 𝜔  stable mode with damped oscillations The most dangerous coupled mode when 𝜔 𝑞 ≈0 𝜇=−𝑞𝑀− 𝑄 0 −1=73504 Robust feedback for instability suppression

RW - Longitudinal coupled bunch instability Growth rate for the 𝜇-th mode 1 𝜏 𝜇, ∥ = 𝑒𝜂 𝐼 𝑏 4𝜋𝐸 𝑄 𝑠 𝑞 𝜔 𝑞 𝑅𝑒 𝑍 ∥ 𝜔 𝑞 𝐺 ∥ 𝜎 𝑧 𝑐 𝜔 𝑞 ′ =1 with 𝜔 𝑞 = 𝑞𝑀+𝜇+ 𝑄 𝑠 𝜔 0 𝑅𝑒 𝑍 ∥ 𝜔 𝑞 >0 Stability for negative 𝜔 𝑞 The worst unstable case is 𝜔 𝑟 = 𝜔 𝑞 >0 Coupling with a single frequency line 1 𝜏 𝜇, ∥ = 𝑒𝜂 𝐼 𝑏 4𝜋𝐸 𝑄 𝑠 𝜔 𝑟 𝑅 𝑠 Max 𝑅 𝑠 producing a growth rate compensated by the natural radiation damping HOM shunt impedance

Other impedance sources Other vacuum chamber components (absorbers, RF cavities with tapers) already evaluated Contribution negligible compared to the RW one 4000 Beam Position Monitors 4000 winglet-to-circular double tapers 8000 bellows with RF fingers CST wakefield simulations (Gaussian bunch of 4mm RMS bunch length)

Other impedance sources RW analytic wake potential (no coating case) Sirius BPM seems to be a good candidate Contributions of tapers and bellows with RF fingers not negligible

Conclusions The effects of the resistive wall on the beam dynamics have been analyzed No Beamstrahlung: MI regime, i.e. the beam would be unstable (for NEG MI threshold is below the nominal bunch intensity) Beamstrahlung allows to have stable beams injection with alternating beams would allow a good margin of safety TMCI not dangerous: threshold far beyond the nominal bunch intensity Robust feedback system required for the fast (6 turns) transverse instability suppression Maximum allowed shunt impedance of a HOM as a function of its resonant frequency estimated Other impedance sources have been evaluated SR absorbers and RF cavities with tapers negligible w.r.t. RW The four-button BPM design seems to be a good candidate The contributions of tapers and bellows with RF fingers are comparable with the RW one The impedance can be reduced by Removing the tapers by placing the BPMs on the winglet vacuum chamber Using a different design for the bellows, for example the comb-type RF shield proposed for KEKB.